The feasibility of physics-based forecasting of earthquakes depends on how well models can be calibrated to represent earthquake scenarios given
uncertainties in both models and data. We investigate whether data assimilation can estimate current and future fault states, i.e., slip rate and
shear stress, in the presence of a bias in the friction parameter. We perform state estimation as well as combined state-parameter estimation using
a sequential-importance resampling particle filter in a zero-dimensional (0D) generalization of the Burridge–Knopoff spring–block model with rate-and-state
friction. Minor changes in the friction parameter

Earthquake hazard quantification requires estimates and uncertainties of parameters such as the long-term average recurrence rate of earthquakes. Hence, modeling earthquake sequences may help us to understand and forecast the processes that determine these recurrence intervals. Physics-based models of the fault
are therefore needed to predict the time when a subsequent earthquake will occur (

Frictional parameters are important in the evolution of fault slip, as they largely determine the recurrence interval of earthquakes. Thus, poorly known or misrepresented parameters can introduce a bias, which can be an important source of uncertainty in the model. If this bias is not corrected, the forecasts obtained using the forward model can be misleading. In previous studies, the frictional parameters have either been estimated as part of the data assimilation or assumed to be perfectly known. In this study, we will investigate the ability of state-estimation methods to correctly update the states and compensate for a parameter bias as well as exploring the ability of state-parameter estimation methods to reduce or even remove parameter bias.

The objective of this paper is to evaluate the effectiveness of data assimilation for state estimation and state-parameter estimation in the presence
of a parameter bias. To address this, we consider various cases: one set of cases in which the parameter is assumed to be known but has a biased value
and one set of cases in which the parameter is estimated along with the state. In these cases, we model fault slip across faults separating tectonic
plates using a spring–block slider model, which is assumed to obey a rate- and state-dependent friction formulation

Let the vector

Consider a vector

In the following, we describe a data-assimilation method for a generic state vector

A Monte Carlo representation in the form of particles can be used to approximate the posterior pdf

We refer to these realizations as “particles”, whereas they may be referred to as “ensemble members” in other studies.

When we insert Eq. (

Here, the weights

Thus, the values of the weights are determined by the likelihood

As the number of particles is typically too small to have sufficient samples of the prior, we observe that, as time progresses, most of the particles
obtain negligible weights, whereas one or a few particles obtain a very high weight. This is commonly referred to as “filter degeneracy” (e.g.,

In the present study, the likelihood

In our study, we have implemented a systematic resampling algorithm, as it provides better estimates compared with other resampling methods used in
data assimilation (

A simplified, computationally efficient description of earthquake sequences is provided by a spring–block slider system, often referred to as the
Burridge–Knopoff (BK) model for frictional sliding

We describe the frictional force using a laboratory-derived rate- and state-dependent friction formulation

Here, the capital notation

In this study, we consider a zero-dimensional (0D) version analyzing a single spring–block slider (Fig.

Finally, the internal parameter

When compared to a slip-weakening friction formulation, the parameter

Phase diagram for a Burridge–Knopoff (BK) model with a rate-and-state friction formulation when (1)

Parameters of the reference model.

To study stick–slip behavior, we limit our 0D model to a velocity-weakening regime for which

In state estimation in the seismic-cycle model, the state

By updating this state vector, we update both the state and the parameter at each new analysis time.

In the following, we consider a “truth run” with

As the true state of the fault system is unknown, we validate our data-assimilation algorithms using twin experiments. Using these synthetic
experiments, we can assess whether data-assimilation methods are able to correctly evaluate the posterior distribution of the state. For the synthetic
truth, we select a forward model run that follows Eqs. (

For state and parameter estimation, a different approach has been adopted to generate the prior. As we observe that the state variables are highly
sensitive to the parameter value, we make sure that prior state variables are fully consistent with the prior parameter values. For this, we use
the following procedure:

We sample

For each

We sample the initial conditions of the state variables (

In this way, each state variable and the parameter chosen in the ensemble (

Synthetic observations are produced by sampling from the synthetic truth and adding an observational error from a Gaussian distribution with an SD of

The experiments are performed with 1000 particles using a sequential-importance
resampling (SIR) particle filter. The observations are provided for the two state variables

The evolution of 1000 particles along with the posterior mean, observation and truth of state variable

Figure

Shear-stress evolution. The coseismic and interseismic phases in the seismic cycle have been highlighted. The time points discussed in the study are

Shear-stress evolution with weight distribution at time

Particles in the

The pdfs representing the posterior mean (red), the prior (gray line), the observation (magenta) and the truth (black line) for the case illustrated in Fig.

To better understand and improve these data-assimilation results, we analyze the results for the experiment with intermediate bias in more detail
(Fig.

The particles with a parameter bias show a phase difference compared with the truth (Fig.

The evolution of particles with time for intermediate bias in state estimation with

We note that a biased parameter value makes it physically impossible to have

The model error in the forward model equation represents the imperfections in the model and, thus, maintains ensemble spread. Adding

We find that inclusion of model error has a noticeable positive effect on the posterior estimate (brings the mean of the posterior closer to the truth) when the parameter bias is either small or intermediate.

We observe that increasing the model error causes the prior

Resampling improves the effectiveness of the particle filter by introducing more particles with a state close to the observed state
(Sect.

We study the effect of increasing the model error with consequent double resampling on the posterior estimates and use the term “combined
adjustment” for this data-assimilation setting. The posterior distribution of stress for the combined adjustment captures the truth distinctly
better compared with the case with model error only. There is no significant difference in the forecasting ability of the prior particle distribution
in the first seismic cycle for the combined adjustment. Increasing model error and double resampling both increase the variability within the particles.
However, at time

The

The pdfs for the case with

Figure

The parameter estimate gradually comes close to its true value after a certain time period. In the state-parameter estimation case, the parameter
estimate gradually changes from its prior (biased) value towards the correct estimate of

To evaluate the overall accuracy of the state and the joint state-parameter estimation for the different biased-parameter cases, we perform a
regression analysis. The results in Fig.

Comparison of regression between the estimate of

The results of this study illustrate the implications of parameter bias in data-assimilation applications for seismic-cycle modeling. They show that it is feasible to apply a particle filter to a fault slip model; moreover, they reveal that uncertainties in friction parameters can be accounted for by either directly estimating these parameters or via a larger model error. By accounting for these improved parameter estimates, fault state estimates can also be improved. This demonstrates that a possible trade-off between estimating shear stress and friction can be effectively accounted for using data-assimilation methods. In other words, data assimilation is able to separate error contributions due to uncertainties in friction (i.e., shear over normal stress) from uncertainties in estimates of a fault's shear stresses. This suggests that potentially large uncertainties in friction do not hamper the further development of data-assimilation methods nor physics-based seismic hazard assessment.

The simplest and most effective approach to deal with parameter bias depends on the degree of parameter bias observed. For laboratory experiments
with increased accuracy, one can potentially still use and tune state estimation (e.g., via increasing model error). However, for observed
uncertainties in laboratory experiments such as in

It is important to remember that these findings are based on the performance of state update and state-parameter update algorithms for a simplified
nonlinear physics-based model in synthetic experiments. Some limitations that are still present in this pioneering study, which eventually need to be
addressed, include the following:

In this study, we assimilate both the fault shear-stress and the slip rate observations on the plate interface. However, we have also conducted experiments only assimilating the fault shear-stress observations. In the latter case, it is observed that the state estimate yields better fits to the truth compared with the case in which we assimilate both types of observations or only the slip rate. Assimilating slip rate observations leads to lower weight values owing to their relatively high observation error. This affects the posterior estimate. However, a thorough study still needs to be conducted to identify which observations provide the most meaningful information when assimilated in earthquake-cycle models.

In this study, we focus only on the uncertainty with respect to parameter

Typically, earthquake forecasting is approached in a probabilistic manner

Another point of attention is the selection of the number of particles required for a correct sampling of the prior. On the one hand, increasing
the number of particles can improve estimation accuracy, but limited computational resources can make this impractical. On the other hand, having a
lower number of particles increases the risk of filter degeneracy. In this study, we initially used 50 and 100 particles but increased the number of
particles to 1000 to avoid degeneracy. Realization of 1000 particles may be computationally expensive for models that are more complex than the
model used here. Improved sampling techniques, such as using a proposal density function, or using a particle flow

An additional point worth mentioning is the use of synthetic observations for fault displacement and velocities for data assimilation in this
study. In realistic applications, the assumptions that we have made with respect to data-assimilation frequency and the SD and
distribution of the observational errors may not be valid. However, if we know the distribution of the measurement errors, we can use that
information to choose the relevant likelihood function, which can greatly effect our fault estimates. Fault shear-stress observations are usually
not available or are subject to large errors if they are available. In contrast, fault velocities can be observed fairly accurately using GPS; this has been
discussed by

It is also very important to highlight the reason behind selecting a 0D model for this study. Simplified fault slip models
are computationally efficient tools that help us to understand the physics behind earthquake dynamics. A study by

Nonetheless, simplified models provide useful insights to solve more complex problems. This is an important stepping stone in the development of
data-assimilation applications for the simulation of more realistic earthquake cycles. Future developments for these purposes, addressing the above
limitations, could utilize other methods explicitly accounting for parameter bias in the data-assimilation scheme in order to obtain a better accuracy

In this study, we demonstrated the effect of a parameter bias in an earthquake-cycle model on the estimated states with data assimilation. Synthetic,
noisy observations of the shear stress and slip velocity on the fault plate interface were assimilated with state and state-parameter estimation
methods using a particle filter. In our forward model, the shear-stress estimates strongly depend on the friction parameter

For a small bias in friction parameter

The observations in the study were generated via simulation. Extra explanatory figures have been uploaded to the Supplement.

The supplement related to this article is available online at:

AB framed the theoretical conceptual model, performed the analytic calculations and conducted the numerical simulations. Both FCV and YvD contributed to and supervised the final version of the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The contributions of Femke C. Vossepoel and Arundhuti Banerjee have been funded by the Delft Technology Fellowship of Delft University of Technology. This work contributes to the DeepNL InFocus project, funded by NWO (DEEP.NL.2018.037). The work benefited from discussions with André Niemeijer of Utrecht University and Hamed Diab-Montero of Delft University of Technology.

This research has been supported by the NWO (grant no. DEEP.NL.2018.037).

This paper was edited by Ilya Zaliapin and reviewed by two anonymous referees.